Video summary
The video presents a mathematical puzzle involving a standard chess board from which two opposite corners, specifically the lower left and upper right squares, have been removed. This modification leaves exactly 62 squares on the board instead of the usual 64. The challenge is to determine whether it is possible to cover these remaining 62 squares completely using 31 rectangular tiles, where each tile covers precisely two adjacent squares. At first glance, one might assume that since there are enough tiles to match the number of available spaces (31 tiles times 2 squares per tile equals 62), a solution must exist; however, the video guides viewers through a logical deduction process using color patterns as the key constraint.
To solve this problem, the narrator explains the alternating black and white coloring pattern inherent in any standard chess board, noting that there are originally an equal number of each: 32 black squares and 32 white squares. By observing the specific corners removed from the board, it becomes clear that both missing squares share the same color; based on the traditional layout where opposite corners match, these two absent squares must be black. Consequently, after removing them, the modified board contains 30 black squares but retains all 32 white squares, creating an imbalance between the two colors.
The solution hinges on how each rectangular tile interacts with this color pattern. Regardless of whether a rectangle is placed horizontally or vertically to cover two adjacent squares, it will always encompass one black square and one white square because neighboring squares on a chess board are never the same color. Therefore, any arrangement using 31 such tiles would necessarily require covering an equal number of black and white squares—specifically 31 of each. Since the modified board possesses unequal numbers of these colors (30 black versus 32 white), it is mathematically impossible to cover every square without leaving at least two uncolored, proving that no valid arrangement exists for this puzzle.
Read the full video transcript
hello and welcome to the session in this
session we are going to solve a
mathematical
Challenge in a chess board two squares
are missing one the lower left
square and one is the upper right Square
as shown in the
figure that leaves 62 squares on the
board you are given 31 articles each
article is a rectangle that can cover
exactly two squares on the chess
board can you place all 31 articles on
the chess board so that all 62 squares
are
covered before solving this challenge
let me give you a hint which can help
you in solving this
challenge this challenge
is based on
squares
rectangles and pattern of
colors for this you must know the shape
of a square and a
rectangle see these two shapes here all
four
sides are equal
and all
angles are at
90° so they both are
squares now if I join them
together like this then I have a
rectangle thus according to this
challenge two squares will together form
a
rectangle also let us understand the
term pattern a
pattern is a repetitive
design for example example see the color
of balls in this
figure here one ball is colored
orange and other is
colored
blue
alternatively so they are forming a
pattern of
orange and blue colored
balls so using these hints try and
complete the challenge observe the
statement carefully you may pause the
video to try it yourself
were you able to solve it let me help
you in this challenge we are given a
chess
board 2 squares have been removed from
this chest board one is the upper right
square and the other is the lower left
Square initially we had 64
squares and now after removal of two
squares we are left with 62
squares now the challenge says that we
have 31 rectangular
articles that can exactly cover two
squares on the chest
board we have to tell whether we can
place these 31 articles on the chest
board so that all 62 squares are covered
[Applause]
now see at first instance we may see
that there are 31
rectangles each rectangle covers two
squares so total squares covered will be
31 into 2 which is equal to
62 so all squares are covered
but it is not the case we have to see
the pattern formed in the
chessboard here in the figure we see
that squares on the chess board are
colored black and white
alternatively one is White and the other
is black then again White
and black so there is a pattern formed
by the black and white
colors so our first step is to
count the number
of black and white squares
now we know on a chessboard there are 64
squares half of them are black and half
of them are
white so we have 32 black
squares and 32 white
squares on a chest
board now since we have two squares
missing from the chest board let us now
find squares of which color have been
removed according to the
pattern this upper right Square should
be of black
color also this lower left Square should
also be of black color
color so we observe that both missing
colors are of black color so now we have
32 white
squares and only 30 black
squares on the chest board now our
second step is to see
how the
rectangle can cover any two
squares see if we try to place the
rectangle on any two squares
we can either place it
vertically or
horizontally if we place it vertically
then out of the two squares covered one
will be black and one will be
white similarly if we place it
horizontally then also one black and
white square will be
covered it means if we have 31
rectangles covering 62
squares we must have same number of
black squares and same number of white
squares it means we must have
31 white
squares and 31 black Square
is but since we have removed two black
squares we are left
with 30 black
squares and 32 white squares
so the given 31 rectangular articles
cannot cover the given chess
board wasn't that easy so with this we
have completed our challenge hope you
enjoyed it